Center of gravity in a non uniform gravitational field

[luisgml_2000]

Hi:

I have a very big concern about my mechanics homework because a problem says: "Find a formula that gives the position of the center of gravity and make sure that in the limiting case of a uniform gravitational field your formula predicts that both center of gravity and center of mass coincide"

 

[jdavel]

luisgml,

Since you don't say what your "very big concern" is, I'll just guess: you don't understant what they're asking for.

Take a simple object - two point masses, 1kg and 2kg connected by a massless, rigid rod 1m long. Where is the center of mass?

Now suppose the gravitational field at the location of the 1kg object is twice as strong as it is at the location of the 2kg object (assume the direction of the field is the same at both locations). What seems like a reasonable definition for the "center of gravity", and where will it be?

 

[GSXR750]

Hello,

I can understand your concern about this problem. The center of gravity is the point at which the entire weight of an object can be considered to act upon. In a uniform gravitational field, the center of gravity and the center of mass will coincide because the gravitational force is constant throughout the object.

However, in a non-uniform gravitational field, the gravitational force will vary at different points on the object. This means that the center of gravity will also vary and will not necessarily coincide with the center of mass.

To find the position of the center of gravity in a non-uniform gravitational field, we can use the following formula:

x = ∫xρ(x)dV / ∫ρ(x)dV

Where x is the distance from a chosen reference point, ρ(x) is the density at point x, and dV is the differential volume element. This formula takes into account the varying density of the object in a non-uniform gravitational field.

To ensure that this formula predicts the same result as the center of mass in a uniform gravitational field, we can set the density to be constant throughout the object (ρ(x) = ρ) and integrate over the entire volume of the object. This will result in a constant value of x, which is the location of the center of mass.

I hope this helps clarify the concept of center of gravity in a non-uniform gravitational field. Let me know if you have any further questions. Good luck with your homework!